Optimal. Leaf size=22 \[ \log \left (x+x^2+\log \left (1+\left (2-e^{e^x}\right ) (-2+x)\right )\right ) \]
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Rubi [F] time = 4.34, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1+4 x-4 x^2+e^{e^x} \left (-1+e^x (-2+x)-3 x+2 x^2\right )}{3 x+x^2-2 x^3+e^{e^x} \left (-2 x-x^2+x^3\right )+\left (3+e^{e^x} (-2+x)-2 x\right ) \log \left (-3+e^{e^x} (2-x)+2 x\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1+4 x-4 x^2+e^{e^x} \left (-1+e^x (-2+x)-3 x+2 x^2\right )}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx\\ &=\int \left (\frac {1}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )}-\frac {e^{e^x}}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )}+\frac {e^{e^x+x} (-2+x)}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )}+\frac {4 x}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )}-\frac {3 e^{e^x} x}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )}-\frac {4 x^2}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )}+\frac {2 e^{e^x} x^2}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )}\right ) \, dx\\ &=2 \int \frac {e^{e^x} x^2}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx-3 \int \frac {e^{e^x} x}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx+4 \int \frac {x}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx-4 \int \frac {x^2}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx+\int \frac {1}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx-\int \frac {e^{e^x}}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx+\int \frac {e^{e^x+x} (-2+x)}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx\\ &=2 \int \frac {e^{e^x} x^2}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx-3 \int \frac {e^{e^x} x}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx+4 \int \frac {x}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx-4 \int \frac {x^2}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx+\int \frac {1}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx-\int \frac {e^{e^x}}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx+\int \left (-\frac {2 e^{e^x+x}}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )}+\frac {e^{e^x+x} x}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {e^{e^x+x}}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx\right )+2 \int \frac {e^{e^x} x^2}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx-3 \int \frac {e^{e^x} x}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx+4 \int \frac {x}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx-4 \int \frac {x^2}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx+\int \frac {1}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx-\int \frac {e^{e^x}}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx+\int \frac {e^{e^x+x} x}{\left (3-2 e^{e^x}-2 x+e^{e^x} x\right ) \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.94, size = 22, normalized size = 1.00 \begin {gather*} \log \left (x+x^2+\log \left (-3-e^{e^x} (-2+x)+2 x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 20, normalized size = 0.91 \begin {gather*} \log \left (x^{2} + x + \log \left (-{\left (x - 2\right )} e^{\left (e^{x}\right )} + 2 \, x - 3\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 37, normalized size = 1.68 \begin {gather*} \log \left (x^{2} + x + \log \left (-{\left (x e^{\left (x + e^{x}\right )} - 2 \, x e^{x} - 2 \, e^{\left (x + e^{x}\right )} + 3 \, e^{x}\right )} e^{\left (-x\right )}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 22, normalized size = 1.00
method | result | size |
risch | \(\ln \left (x^{2}+x +\ln \left (\left (2-x \right ) {\mathrm e}^{{\mathrm e}^{x}}+2 x -3\right )\right )\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 20, normalized size = 0.91 \begin {gather*} \log \left (x^{2} + x + \log \left (-{\left (x - 2\right )} e^{\left (e^{x}\right )} + 2 \, x - 3\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {4\,x-{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (3\,x-{\mathrm {e}}^x\,\left (x-2\right )-2\,x^2+1\right )-4\,x^2+1}{3\,x+\ln \left (2\,x-{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (x-2\right )-3\right )\,\left ({\mathrm {e}}^{{\mathrm {e}}^x}\,\left (x-2\right )-2\,x+3\right )+x^2-2\,x^3-{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (-x^3+x^2+2\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.88, size = 20, normalized size = 0.91 \begin {gather*} \log {\left (x^{2} + x + \log {\left (2 x + \left (2 - x\right ) e^{e^{x}} - 3 \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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